Geometric Origin of Gauge Coupling and Unification of Four Forces in Memory Theory: A Falsifiable Path from D_8 Symmetric Vacuum to Standard Model Gauge Group

The series works of Memory Theory propose that gravity is a density-dependent restoring force of the vacuum responding to matter perturbation, with its core index \alpha = \sin(\pi/8) originating from the D_8 octagonal symmetry of vacuum geometry. Based on this framework, this paper proposes a geometric origin hypothesis for gauge interactions: the Standard Model gauge group SU(3)_c \times SU(2)_L \times U(1)_Y is an effective low-energy description emerging from D_8 symmetry through subgroup stratification, spontaneous symmetry breaking and continuum limit. We present the corresponding rules from C_4 \to U(1)_Y, C_2 \times C_2 \to SU(2)_L, and three fractal branches of D_8 \to SU(3)_c. This paper further redefines the relationship between gravity and gauge interactions as a physical stratification of potential energy and kinetic energy: gravity is the potential energy response (classical) of vacuum geometry, while electromagnetic, strong and weak forces are quantum excitation modes above this potential. The scale dependence of gauge coupling constants is interpreted as the scaling behavior of density-dependent restoring force in Memory Theory: \rho(Q) \propto Q^4, \mu^2(\rho) \propto \rho^\alpha, leading to g^2(Q) \propto Q^{4\alpha}, namely power-law running. At the electroweak scale m_Z, the vacuum satisfies the geometric resonance condition, and the strong coupling constant is precisely locked as \alpha_s(m_Z) = \frac{11}{2}\sin^4(\pi/8), with an error less than 0.05\% compared with current experimental values. This paper also gives testable predictions such as black hole shadow offset and scalar polarization of gravitational waves, and systematically discusses the emergent path from D_8 to gauge groups. All core predictions can be falsified by experiments within the next 10–20 years.